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Ann and Bea leave Townville at the same time and travel towards Villag
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25 Apr 2021, 11:52
25 Apr 2021, 11:52
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Ann and Bea leave Townville at the same time and travel towards Villageton, which is 2K kilometers away. Their individual speeds are constant, but Ann’s speed is four times Bea’s speed. Upon reaching Villageton, Ann immediately turns around and drives toward Townville until she meets Bea. When they meet, how many kilometers has Bea traveled?
A) K/5
B) K/4
C) 2K/3
D) 3K/4
E) 4K/5
A) K/5
B) K/4
C) 2K/3
D) 3K/4
E) 4K/5
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Joined: 10 Apr 2015
Posts: 6218
Given Kudos: 136
Re: Ann and Bea leave Townville at the same time and travel towards Villag
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25 Apr 2021, 11:59
25 Apr 2021, 11:59
1
Carcass wrote:
Ann and Bea leave Townville at the same time and travel towards Villageton, which is 2K kilometers away. Their individual speeds are constant, but Ann’s speed is four times Bea’s speed. Upon reaching Villageton, Ann immediately turns around and drives toward Townville until she meets Bea. When they meet, how many kilometers has Bea traveled?
A) K/5
B) K/4
C) 2K/3
D) 3K/4
E) 4K/5
A) K/5
B) K/4
C) 2K/3
D) 3K/4
E) 4K/5
Here's an algebraic solution:
Let B = the distance Bea traveled
Let R = Bea's speed.
NOTE: the total distance from Townville to Villageton and then BACK TO Townville = 4K.
So, 4K - B = the distance Ann traveled
And 4R = Ann's speed (since her speed is 4 times Bea's speed)
From here, let's create a WORD EQUATION that uses distance and speed.
How about: Ann's travel time = Bea's travel time
Time = distance/rate, so we get:
(4K - B)/4R = B/R
Cross multiply to get: (B)(4R) = (R)(4K - B)
Expand: 4BR = 4RK - BR
Add BR to both sides: 5BR = 4RK
Divide both sides by R to get: 5B = 4K
Divide both sides by 5 to get: B = 4K/5
Answer: E
General Discussion
Retired Moderator
Joined: 10 Apr 2015
Posts: 6218
Given Kudos: 136
Re: Ann and Bea leave Townville at the same time and travel towards Villag
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21 Oct 2022, 15:31
21 Oct 2022, 15:31
1
Carcass wrote:
Ann and Bea leave Townville at the same time and travel towards Villageton, which is 2K kilometers away. Their individual speeds are constant, but Ann’s speed is four times Bea’s speed. Upon reaching Villageton, Ann immediately turns around and drives toward Townville until she meets Bea. When they meet, how many kilometers has Bea traveled?
A) K/5
B) K/4
C) 2K/3
D) 3K/4
E) 4K/5
A) K/5
B) K/4
C) 2K/3
D) 3K/4
E) 4K/5
A student asked me to solve this question by choosing numbers (aka the INPUT-OUTPUT method).
So let's do that.
Given: Ann’s speed is four times Bea’s speed
So, let's say Bea's speed is 10 kph, and Ann's speed is 40 kph.
Given: Distance from Townville to Villageton = 2K kilometers
So, let's say K = 100 km, which means the distance from Townville to Villageton = 200 km
Let's first examine 200 km Ann's trip Townville to Villageton.
time = distance/rate
So, the time it takes Ann to reach Villageton = (200 km)/(40 kph) = 5 hours
During the 5 hours it takes Ann to reach Villageton, Bea travels a distance of 50 km.
We know this because distance = (rate)(time)
So the distance Bea travelled = (10 kph)(5 hours) = 50 km
So, after 5 hours, when Ann turns around to head to back to Townville, she and Bea are 150 km apart.
So, now let's calculate how long it will take for the 150 km gap between Ann and Bea to shrink to 0 km (i e, when they meet)
Since Ann and Bea are now headed towards each other, this is a shrinking gap question.
The rate at which the gap shrinks = the sum of their two speeds
10 kph + 40 kph = 50 kph, which means the gap shrinks at a rate of 50 kph
So, the time it takes for the 150 km gap between Ann and Bea to shrink to 0 km = distance/speed = (150 km)/(50 kph) = 3 hours.
So, the Ann's TOTAL travel time = 5 hrs (the time it took to reach Villageton) + 3 hours (the time it took to meet Bea on her way back to Townville) = 8 hrs
Since Ann and Bea must have the same travel time, we know that Bea's total travel time also equals 8 hrs
The question asks: When they meet, how many kilometers has Bea traveled?
We know that distance = (rate)(time)
So, Bea's travel distance = (10 kmh)(8 hrs) = 80 km
We can now conclude that, when we INPUT K = 100, the answer to the question (aka the OUTPUT) is 80 km
So, now let's plug K = 100 into each answer choice to see which one year olds and output of 80
A) 100/5 = 20. No good; we want an output of 80.
B) 100/4 = 25. No good; we want an output of 80.
C) 2(100)/3 = 66.666.... No good; we want an output of 80.
D) 3(100)/4 = 75. No good; we want an output of 80.
E) 4(100)/5 = 80. BINGO!
Answer: E
As you can see, the input-output approach is a lot of work for this particular question. But it still works!
